PHYS3009: Force and function at the nanoscale
Handbook 2023-24
Module Convenor: Dr Mike Smith, Rm C126, Physics Building, mike.i.smith@nottingham.ac.uk
This is a 10 credit Spring semester module
About the course#
Aims of the Module#
To provide an introduction to the kinematics of nanoscale systems
To help students to develop an understanding of the origin of forces on the nanoscale
To help students develop an appreciation of the role of intermolecular, interparticle and surface forces in nanoscale science, technology and biology.
Assessment#
Examination 100%:
In the summer assessment period.
In person, closed book
2 hrs to complete paper
Access to an equation sheet containing some key equations. This will be available in advance
Contents:#
The Nanoscale World
Forces and potentials
Polar interactions
Dispersion interactions
Hydrogen Bonds & the Hydrophobic Effect
Measurement of nanoscale forces
Surface energy, surface tension and adhesion energy
Capillary pressure
Colloidal interactions
Steric interactions and entropic repulsion
Depletion Interactions
Aggregation
Self-assembly
Using this guide#
The material in this module is drawn from a range of different sources making it difficult to recommend a single textbook that is written at the appropriate level that covers the material. As such I have put together these companion notes which function like a textbook for the course. All the material is covered in the same order as the course. Some students prefer written rather than oral explanations, and sometimes it is quicker and easier to look up / remind yourself about something specific in a sequential written text. You shouldn’t therefore feel you need to read this guide cover to cover, rather use it to complement the rest of your studies, looking up bits that perhaps you are unclear on or need to go over more slowly.
Questions?#
If you are stuck on any part of this course, try reading through the appropriate part here but if you are still struggling to understand, here are the options:
I’m always happy to answer questions during or immediately after lectures.
Every written question I answer from students will be posted to the Q&A forum of the Moodle page, so it is worth having a look there in case your question has already been addressed. If you want to see all questions and answers you may subscribe to this feature.
If your question is not already listed then please email me your question at:
I’m always happy to answer your questions and will usually get back to you in 1-2 working days. I will reply directly to your question and post in an anonymized way with the corresponding answer. I prefer to do this since then your question can then benefit others.
1. The Nanoscale World#
The Nanoscale World#
The nanoscale world is very different from the macroscopic one that we inhabit. Thermal fluctuations constantly buffet and agitate molecules so that a good mental picture is one of ceaseless random motions. Bringing some order to this chaos, nanoscale forces attempt to resist these disruptive influences.
Microscopic and Nanoscale Forces#
Interatomic, intermolecular and surface forces act between different objects and are responsible for holding (or not holding) materials such as solids, liquids and gases together. These forces often have a simple power law dependence, but they are all extremely short-range interactions that persist over distances of 0.2 nm to 100nm.
Brownian motion and what it tells us about the micro / nanoscale world#
Unlike the macroscopic world, the motion of atoms, molecules and sub-micron particles is dominated by the thermal fluctuations of atoms and molecules that surround them. Each object is constantly being pummeled by atoms and molecules in their environment which are in turn being pummeled by their neighbours. This means that motion through a liquid on the nanoscale is more like trying to swim through a children’s ball pool (where the balls are constantly being fired at you from all sides using a tennis ball cannon) and less like trying to swim as one might in a swimming pool full of water.
My point here is that the constant thermal motion that occurs on the sub-micron and nanoscale makes motion very different on these length scales (and subsequently has significant consequence for the design of swimming objects on these length scales – although this will not be covered in this module). Instead of smooth fluid like motion, like that obtained in bulk liquids (where inertial effects are important), we find that the atoms, molecules and small particles diffuse around, executing a random walk such that their mean square displacement <x2> after a time, t, is given by
 [equation 1.1]
i.e. their displacement varies as t0.5.
The constant D is referred to as the diffusion coefficient (m2s-1) and can be related to the size of the diffusing object and the viscosity,, (Pas) of the surrounding medium. For a spherical object of radius a this is given using the Stokes-Einstein equation  [1.2]
The thermal energy scale#
At temperatures above absolute zero the atoms can have a significant amount of thermal energy. This thermal energy causes the atoms and molecules in a substance to jiggle around constantly, bashing into one another with effect of disrupting any interactions between the atoms. When the thermal energy associated with atoms, molecules and particles becomes comparable to the energy of interactions, the interactions will cease to be significant.
We therefore need to determine the magnitude of the thermal energy scale and to determine how this influences forces (e.g. Gravity and Van der Waals).
Suppose we start by considering the atoms/molecules in an ideal gas whose behavior can be described using the ideal gas equation.
 [1.3]
This equation relates the pressure P (Nm-2) and volume V (m3) of a gas, to the number of moles, n, and the temperature, T (Kelvin) using the molar gas constant R (Jmol-1K-1). It is an equation with which you should be familiar and which you may have encountered at A-level (if not in your first year).
The left-hand side of this equation has units of energy (Nm-2 x m3 = Nm)
In fact, PV is the total internal energy of the gas, which is manifested as the kinetic (or movement) energy of the molecules.
As both sides of the equation must have identical units, the right-hand side must also have dimensions of energy. If we note that the number of moles of a substance, n, is simply the number of atoms/molecules in the gas, N, divided by Avogadro’s number, NA (=6 x 1023) we can write
 [1.4]
So the total internal energy of the gas is NkT, where k is Boltzmann’s constant (k=1.38 x 10-23 JK-1). This means that the average thermal energy associated with each of the N atoms in the gas is
 [1.5]
In fact, things are not quite this simple as the internal energy of a gas depends upon the properties of the individual molecules (cf thermal physics module). However, this very simple result does provide the correct order of magnitude for the thermal energy scale at a given temperature T.
How big is kT?#
At room temperature, T=300 K, so kT = (1.38 x 10-23)(300) = 4.14 x 10-21 J (or 25 meV).
For two Neon atoms at a separation of 0.2 nm, their Van der Waals energy of interaction Udispersion is given by
 (or 38 meV)
These two energies are very similar therefore we expect that the Van der Waals interactions between individual atoms and molecules would be disrupted by their constant thermal jiggling on length scales larger than a fraction of a nanometre. This turns out to be the case, confirming that the thermal energy scale is important on the nanoscale
The Boltzmann Equation#
The Boltzmann equation enables us to compare the importance of the thermal fluctuations with energy (kBT) to the strength of a particular interaction or potential. This gives probability that thermal energy disrupts an interaction of strength U. N.B it is only proportional to not equal to
[1.6]
In the lectures we thought about some particles suspended in a liquid. Depending on their size they either sedimented or stayed suspended in the liquid. We can use the Boltzmann probability to understand this. That is that for small enough particles, gravity is of negligible importance. The gravitational potential of a particle is given by mgz where z is the height mg is the effective weight (ie weight of particle minus the upthrust of the liquid). We can therefore rewrite the Boltzmann probability in terms of the concentration of particles at different heights.
[1.7]
φ is the volume fraction of particles, that is the number of particles multiplied by the volume of a particle divided by the volume of the containing liquid. If gravity is insignificant then the concentration of particles should change on a lengthscale (lg) much larger than the height of the container.
[1.8]
As we showed at particle radii ≤ 1µm particles are dominated by thermal motions and gravity becomes unimportant. This is fundamental to understanding the nanoscale world. Our macroscopic experience of gravity gives meaning to concepts like up and down. Inertia and momentum, so crucial to Newton’s laws, are likewise no longer important concepts at the nanoscale. We therefore have to do some rethinking…
2. Forces and Potentials#
Relationship between force and potential#
The force, F, on an object can be related to the gradient in its potential energy, U, with respect to distance, x, by the formula
 equation 2.1
This is an extremely important formula in physics and we will use it a lot in this module.
The ‘sign’ of the potential energy
When the potential energy is negative the interaction between two bodies is favourable. Consider the example of the potential energy of a system of positive and negative charges
Potential energy U < 0
Similarly, when the potential energy is positive, interactions are unfavourable.
Potential energy U > 0
The sign of the force
In the conventions used throughout this module positive forces are repulsive and negative forces are attractive. Again consider the electrostatic forces acting between charges of the same and different sign
|F| > 0 |F| < 0
Strong Inter-atomic and intermolecular forces
Electrostatic (Coulomb) Interactions
You have already met Coulomb interactions in the first year ‘Newton to Einstein’ module.
Some atoms and molecules (for example proteins) may acquire a net charge and will interact via electrostatic forces
Figure 2.1 Protein molecules such as Insulin (pictured here) become charged when placed in aqueous environments. The charge that they acquire is sensitive to pH and salt concentrations. These molecules can interact via electrostatic forces
The electrostatic potential energy of two molecules having charges q1 and q2 that are separated by a distance, x, is given by
Equation 2.2
o=8.85 x 10-12 Fm-1
=relative permittivity of medium in which the molecules are suspended
The electrostatic force acting between two molecules having charges q1 and q2 that are separated by a distance, x, is given by
Equation 2.3
The sign of the force depends upon the signs of the charges
If q1q2 < 0 force is attractive (negative sign)
If q1q2 > 0 force is repulsive (positive sign)
The range of interactions
As discussed previously, the thermal motion of atoms and molecules tends to disrupt the interactions between them.
At ‘small’ distances interactions between atoms and molecules are strong enough to overcome the effects of thermal motion.
At ‘large’ distances interactions between atoms and molecules become weaker and are unable to overcome the effects of thermal motion.
Small and large are relative terms and depend on the nature and strength of the interaction between two atoms/molecules.
We can estimate the range of an interaction by comparing the thermal energy to the size (magnitude) of the potential energy of two molecules.
Equation 2.4
When |U(x)| £ kT thermal motion disrupts the interactions
When |U(x)| > kT the atoms/molecules still feel the interactions between them
When the two are equal we see a cross over from one regime to another. The distance, x, at which this occurs gives a measure of the range of an interaction.
Example: Range of electrostatic interactions
We can obtain an estimate of the range Xrange of the electrostatic interaction between two charged atoms/molecules if we equate kT with our expression for the electrostatic potential energy (equation 2.2)
Equation 2.5
Rearranging gives
Equation 2.6
For x ≥ Xrange electrostatic interactions become unimportant relative to thermal energies
x < Xrange electrostatic interactions start to dominate interactions between atoms/molecules
Ionic Crystals
Electrostatic interactions are responsible for the formation of ionic crystals
For example, consider NaCl
Large green ions are Cl-
Small red ions are Na+
Each Cl- ion has
6 nearest neighbour Na+ ions at a distance r
12 next-nearest neighbour Cl- ions at √2_r_
8 next-next-nearest neighbour Na+ ions at √3_r_
We can use this information to calculate the (cohesive) energy per ion holding the crystal together by summing up the contributions to the potential energy from all neighbours to give
equation 2.7
Where α is called the Madelung constant (α=1.748 for NaCl) and e is the electronic charge
(1.6 x 10-19C)
Covalent Bonds
Covalent bonds are highly directional and are chemical in origin (e.g. pi bond created by overlap of d electron orbitals)
These interactions originate from the sharing of valence electrons to facilitate the filling of electronic shells within atoms → stable structures.
The directionality of the bonds is caused by the mutual repulsion of the electrons in neighbouring bond orbitals.
Covalent forces (bonds) operate over short distances 0.1-0.2 nm
The energies associated with the formation of these bonds are typically 2 to 8 eV
You will learn more about these in the 3rd year Solid State module
Metallic bonding
Sharing of electrons between metal atoms gives rise to another form of bonding. However, in metallic bonds, the electrons become delocalised throughout the material in such a way that they are shared between many nuclei .
The ‘sea’ of delocalised electrons helps to screen the repulsion between neighbouring nuclei and hold them together.
Metallic bonds are comparable in strength to covalent bonds (a few eV per bond)
The delocalisation of electrons can be used to explain many of the properties of metals
Optical – The optical properties of metals are determined by how the ‘sea’ of electrons responds to the oscillating electric and magnetic fields associated with incident light
Electrical and Thermal – Delocalised electrons are easier to move, so metals are good conductors of heat and electricity
The bulk properties of gold are determined by how its electrons move around. When we make gold nanoparticles the motion of the electrons becomes confined and the properties change. In the gold nanoparticle solutions shown, the optical properties (colour) are changed by changing the size of the particles.
What do we mean by a “Strong Interaction”?
“Strong” and “Weak” are relative terms. In the context of interactions we can think of a “Strong” bond as something which is stable against thermal fluctuations. That is in a large sample, even if we wait a long time, none of the bonds will be disrupted. In contrast a “Weak” interaction is one which has a reasonable probability of being disrupted and hence the two atoms / molecules being held together by the interaction would then be free to diffuse away from one another.
To define this probability more carefully we return to the Boltzmann probability:
The probability of a bond / interaction being disrupted by thermal fluctuations is proportional to the potential of the interaction divided by kT (N.B it is proportional because the prefactor in front of the exponential is unknown – related to collision probabilities etc). The stronger the interaction the larger the value of U/kT. To illustrate, a covalent bond ~ 100kT whereas a Van der Waal’s interaction (to be covered in later lectures) ~ 2kT.
U/kT ~ 100. à e-100 = 3.7x10-44 à Stable or “strong” interaction
U/kT ~ 2 à e-2 = 0.13 à Probability is significant, meaning the interaction is “weak”
3. Polar interactions and Dipoles
Polar Molecules
Many molecules are electrically neutral and carry no net charge. However, they can still possess an electric dipole moment
Consider the molecule HCl (hydrogen chloride), although it is electrically neutral, the electrons in the HCl bond lie closer to the chlorine nucleus, giving the Cl end of the molecule a slightly negative charge and the H end of the molecule a slightly positive charge of the same magnitude.
The reason for this is that electronegative atoms such as chlorine have a higher affinity for electrons than atoms such as hydrogen. In the case of HCl, the electrons are pulled towards the chlorine atom giving rise to a permanent dipole.
Zwitterions and dipolar ions
In some cases, the polar properties of the molecules depend upon the local environment (e.g. presence of solvents). Water soluble dipolar molecules of this type are called Zwitterions (Zwitter- German meaning ‘hybrid’ )
Some molecules contain both a net charge and a dipole moment. These molecules are referred to as dipolar ions
How do we define an electric dipole moment?
Recall from electromagnetism that a dipole consists of two charges (+q and -q) separated by a distance d
The dipole moment, p, is then defined as
The electric dipole moment has units of Cm.
Electric Field due to a dipole
When calculating the forces that are exerted during dipole-dipole interactions it is first necessary to calculate the field due to an electric dipole.
Consider two charges separated by a small distance, d in a medium of dielectric constant,
We can calculate the field at the point, x, by summing up the fields due to the two point charges. When x >> d this gives the result (for an axial position)
Potential energy of a dipole in an electric field
Consider a dipole in an external electric field. There is no net force on the dipole as the forces on each of the charges are equal and opposite. However, the forces exerted on these charges will create a torque on the dipole and will cause it to rotate.
We can calculate the potential energy (U) of the dipole by calculating the work done (W) in rotating the dipole, as U=-W.
This gives the result:
Polarisation of atoms and molecules using external fields
When an electric field is applied to a neutral atom or molecule, the forces on the charged particles (electrons and nuclei) which comprise the atom or molecule cause some charge separation and hence lead to the formation of an induced dipole on the atom/molecule.
The dipole moment of an atom or molecule can be related to the external applied electric field by
where is the polarisability of the atom/molecule in units C2m2J-1
Polarisability of atoms, molecules and small particles
We can determine the polarisability of an atom, molecule or small particle in an applied external field by balancing the electrostatic forces between the separated charges and the force exerted due to the application external field. This gives the result
Where d is typically smaller than, but comparable to radius, R, of the atom, molecule or particle_._ So scales with the volume of the atom/molecule ( ~ R3 ~ Volume).
Polarisability scales with the volume of the molecule.
Summary of key concepts
Dipole moment
Electric field due to a dipole
Potential energy of a dipole in an E field
Dipole induced by an E field
Polarisability
Dispersion Interactions
Dispersion interactions are always present and act between all atoms and molecules. This is in contrast to other types of interaction such as ionic and covalent interactions, which may or may not be present depending upon the type of atoms/ions/molecules in a material.
Dispersion interactions are dipole-dipole interactions, which arise as the result of temporary local fluctuations in charge density on atoms/molecules.
Temporary fluctuations in charge density create an instantaneous dipole on one atom/molecule.
The resulting electric field produced by this dipole then induces a dipole in a neighbouring atom/molecule
A force is exerted between the two correlated dipoles.
Properties of dispersion forces
Can be attractive or repulsive (usually former)
Tend to have a weak orientational effect on neighbouring atoms/molecules
Can be relatively long ranged between macroscopic bodies (0.2 nm to >10 nm)
They play a role in a many important phenomena such as adhesion, surface tension, wetting, flocculation and aggregation of particles.
Potential energy due to dispersion interactions
The potential energy of the dispersion interaction (and indeed any interaction) is a useful tool in helping us to determine the force between two nanoscale objects, as the force is simply related to the spatial derivative of the potential. We can determine the potential energy of an induced dipole in the E-field due to an instantaneous dipole by using the concepts relating to polar interactions developed already.
Consider an instantaneous dipole of magnitude p1=qd on an atom/molecule at the origin (where q is the magnitude of the charge and d is the separation between the positive and negative charges in the resulting dipole).
If a second atom having polarisability, , is placed at a distance, x, from the origin, this atom will experience a separation of charge and will have an induced dipole moment,
However, we have already shown that the electric field due to the first dipole can be approximated to
Also, we know that the potential energy of a dipole in an electric field is given by
Combining these three results gives the potential energy of the induced dipole in the electric field due to the instantaneous dipole as
where C is a constant (Jm6) which determines the ‘strength’ of the interaction between the two atoms.
This expression is the potential energy due to the dispersion interaction between two atoms, but is equally valid for the dispersion interaction between molecules.
The key point is that the potential energy due to dispersion interactions has a power law dependence which decays as 1/x6 These interactions therefore decay much faster than for example the coulomb interaction which has a much longer ranged 1/x2 dependence.
Dispersion Forces
Having derived an expression for the dispersion interaction potential energy, we can determine the force exerted on each of the two atoms by calculating the derivative of the potential, such that
The Range of Dispersion interactions
When the dispersion potential energy becomes comparable to thermal energies, these interactions will start to become less important. We can calculate the range of the dispersion interaction at a given temperature, T, by equating the dispersion potential energy with the thermal energy.
Rearranging this expression gives
Inserting typical values of =9 x 10-40 C2m2J-1 and p=qd=5 x 10-30 Cm for individual atoms at T=300 K gives a value of around 0.3 nm for the range of interaction.
This distance is comparable to the radius an atom (~ 0.1 nm) and shows that dispersion interactions between atoms, while being short range, are strong enough to hold atoms in close contact against the effects of thermal agitation. These types of interactions are responsible for holding simple liquids and some organic solids together.
Why don’t atoms and molecules collapse in on one another?
The form of the dispersion potential predicts that interactions get stronger as atoms and molecules get closer together i.e. the forces become more attractive the closer they get to one another.
So what prevents atoms and molecules from collapsing into one another? The simple answer is that there must be an even shorter range repulsive interaction which exerts a larger force than that generated by dispersion forces when the atoms and molecules come into close contact.
This short-range interaction arises due to the Pauli exclusion principle:
“Two or more electrons may not have the same quantum state.”
In other words, overlap of electron clouds of two atoms is strongly repulsive. However, this is not something that we can derive from first principles so instead we use an empirical potential of the form,
The Lennard-Jones (6-12) potential
The total interatomic/molecular potential between two atoms or molecules is therefore given by summing the attractive and repulsive potentials, as discovered by Prof. Lennard-Jones,
where B, C, and are all constants.
The plot below shows the shape of the attractive, repulsive and total (attractive + repulsive) potentials as a function of the separation between the atoms/molecules.
The force exerted on each of the atoms/molecules is given by taking the derivative of the Lennard Jones potential and gives
The form of the force versus the separation between the atoms/molecules is plotted below in reduced units.
The equilibrium separation occurs when F=0 i.e. when the derivative of the potential is zero.
At distances less than ~ 1.1, the net (total) force is positive and therefore repulsive.
At distances greater than 1.1, the net (total) force is negative and therefore attractive.
Dispersion interactions between extended bodies
We have considered the dispersion interaction between isolated atoms and molecules. From here we can start to work out how dispersion interactions affect the forces that are exerted on extended and macroscopic bodies that are made up of many atoms/molecules.
Recall that the dispersion potential energy between isolated atoms/molecules has the form
This power law dependence of the potential does not hold for macroscopic objects that contain many atoms/molecules.
So what happens when many atoms/molecules come together? The answer to this question is relatively simple.
If the interaction energy between an atom/molecule and one of its nearest neighbours is given by U, then an atom that is surrounded by N nearest neighbours will have a potential energy of order NU.
In reality, it is not always quite this simple, but this argument is correct in the sense that the more atoms/molecules we have in a cluster, the greater the magnitude of the total potential energy associated with the interactions will be. This means that by bringing many atoms/molecules together we can increase the cohesive energy of the resulting cluster.
Increasing the cluster size only works up to a point, as next nearest neighbours and next-next nearest neighbours experience an increasingly weaker interaction with the central atom/molecule. However, these effects can be strong enough to hold some liquids and some organic solids together, particularly if the molecules that comprise these materials are large and thus able to experience many interactions with neighbouring molecules.
A classic example of molecules held together by dispersion forces are the linear alkanes. These are linear chains of carbon atoms with single bonds between them. Each carbon atom in the chain has two hydrogen atoms attached to it (except at the ends, where each carbon atom has three hydrogen atoms attached to it). The lower alkanes such as methane (CH4) and ethane (C2H6) are gases. However, as the length of the alkane chain is increased, these materials form liquids such as octane (C8H18) and eventually solids such as C18H36. In each case the number of interactions between neighbouring molecules determines their cohesive energy and hence the physical state of the material.
5. Dispersion Interactions – larger objects
Gecko feet
The additive nature of interactions is expected to effect the strength of the dispersion interaction between macroscopic bodies.
Geckos live in hot dusty climates. These creatures have the extraordinary ability that they can climb sheer surfaces and walk upside down across smooth glass plates. The mechanism by which they do this depends on small soft structures on their feet which allow them to stick to almost any surface. These small structures deform when the gecko puts its foot on a surface. This increases the amount of contact between the foot and the surface and thus increases the effective strength of the dispersion interactions between the gecko and the thing it is standing on.
Dispersion interactions between macroscopic bodies
We have seen that the additive nature of short-range dispersion interactions can make these interactions significant, but how can we describe these interactions mathematically?
We will begin by considering the interaction between an isolated atom/molecule and a semi-infinite solid surface separated by a distance D.
Semi-infinite
solid surface
Atom/molecule
Consider a small annulus or ring in the solid surface which has a radius, y, and which is a distance, x, from the atom/molecule
All the elemental volumes in this ring are at the same distance r from the isolated atom, they therefore have the same interaction. Integrating over x and y we can add up all the contributions in the slab (see lectures for details).
Interactions between a slab of material and another semi-infinite slab**.**
Derive an expression for the dispersion forces acting between a perfectly flat slab of area S and a semi-infinite solid if the number density of atoms in the slab and solid are n1 and n2 respectively.
To solve this problem we start with the expression for the potential energy due to dispersion interactions between a molecule and semi-infinite surface which are separated by a distance x.
We then work out the number of atoms in a slice (red in the diagram above). We add up all the contributions to the dispersion interaction between this slice and the semi-infinite surface. Finally, we add up all the slices in the slab to reach the total interaction with the solid surface.
Interactions between a sphere and a semi-infinite slab
Derive an expression for the dispersion forces which act between a sphere of radius R (for separations D>>R) and a semi-infinite solid if the number density of atoms in the 
slab and sphere are n1 and n2 respectively.
As in the last example, we start with the expression for the potential energy due to dispersion interactions between an individual atom/molecule and a flat surface and add up the contributions due to all the atoms/molecules in the sphere.
A general strategy
In all the examples above we follow a similar pattern
Look for the symmetries in the system. Eg. With the sphere it is symmetric about the middle.
Split the problem up into elemental volumes (that is those atoms that are all the same distance from the semi-infinite sheet.
Integrate between the limits to add up the different contributions
The Hamaker Constant
The strength of the dispersion interaction between materials is often quantified in terms of the Hamaker constant, A
n1 and n2 are the number densities of atoms/molecules in the two interacting materials (m-3)
C is the ‘strength’ of the interaction between an atom/molecule from one material and an atom/molecule from the other (Jm6)
A12 is the Hamaker constant for materials 1 and 2 (J)
We can rewrite our expressions for the dispersion interaction energy between two macroscopic bodies in terms of the Hamaker constant
Dispersion Interaction energy between a flat slab of area, S, and a semi-infinite solid
Dispersion Interaction energy between a sphere of radius, R, and a semi-infinite solid
We can also write the forces in terms of the Hamaker constant
Dispersion force between a flat slab of area, S, and a semi-infinite solid
Dispersion force between a sphere of radius, R, and a semi-infinite solid
Hence we can determine the strength of the dispersion interaction A12 between two materials by measuring forces acting between two macroscopic bodies. The importance of a Hamaker constant is that we combine a bunch of constants into a single one which represents how two materials interact.
Some other important geometries
The geometry of the macroscopic bodies is important in determining the separation dependence of the dispersion interaction…
Sphere-sphere
Parallel cylinders
Perpendicular cylinders
How big are Hamaker constants?
Values of A typically lie in the range 10-21 to 10-19 Joules (kTroom = 4.41x 10-21 J)
Taken from Surface and Intermolecular forces by J. Israelachvili 3rd edition.
The range of dispersion interactions between macroscopic bodies
We can calculate the range of the interaction by comparing dispersion and thermal energies i.e. |U(Drange)| ~ kT. In reality, this description does not work for truly macroscopic objects, but is valid on the submicron and nanometre length scales where thermal effects are still important in influencing the behaviour of particles and inter-surface forces.
For the interaction between a sphere of radius R and a solid surface we have
Hence
For values of T=300K, R=10 nm and A12=10-19J, Drange ~ 10R = 100 nm
This is significantly longer range than the interaction between individual atoms/molecules (~0.3nm).
The potential for extended bodies depends on a much lower power of R usually 1/R2 or 1/R3 depending on geometry, whereas for individual molecules it is 1/R6. This is why dispersion forces are so important because in material bodies when we add up the individual interactions we find the range over which they are not disrupted by thermal interactions is much longer.
The magnitude of forces between macroscopic bodies
We can also calculate the magnitude of the forces exerted on macroscopic bodies due to dispersion forces. Sticking with the sphere and surface we have
If a nanoparticle has R=10 nm, A12=10-19J
When D=100 nm, F= 3 x 10-14 N ~ 0.03 pN
When D=10 nm, F= 3 x10-12 N ~ 3 pN
But can we measure forces this small? The answer is yes, but we will come to this soon.
We can also calculate the force between a perfectly flat slab and a semi-infinite solid. Recall that the force is given by
If the area of the slab is S=10-4 m2 (1cm2) and A12 = 10-19J. What is the force between the slab and the surface at a separation of 0.3 nm?
The answer is ~20,000 Newtons, This is a very large force!
Summary of key concepts
Dispersion interactions arise due to instantaneous dipole fluctuations and the dipoles these induce in neighbouring atoms/molecules
The potential energy due to attractive dispersion interactions between atoms and molecules has the form
Range of dispersion forces between individual atoms is ~ 0.3 nm
The total interaction (dispersion + hard sphere repulsion) between neutral atoms and molecules can be described using a ‘6-12’ Lennard-Jones potential of the form
Dispersion interactions between macroscopic bodies occur over significantly longer ranges than for isolated atoms/molecules
We can derive expressions for the dispersion interaction energy and forces between macroscopic objects.
Dispersion Interaction energy between a flat slab of area, S, and a semi-infinite solid
Dispersion Interaction energy between a sphere of radius, R, and a semi-infinite solid
Dispersion force between a flat slab of area, S, and a semi-infinite solid
Dispersion force between a sphere of radius, R, and a semi-infinite solid
The strength of the dispersion interaction can be quantified in terms of the Hamaker constant A
Dispersion forces on nanoscale objects can be on the pN to nN scales
6. Water, Hydrogen Bonds & the Hydrophobic Interaction
In this chapter we will study two interactions that deserve special mention in their own right. These are hydrogen bonding interactions and the hydrophobic interaction. These are both extremely important interactions in biology and as we will see in this and future chapters, they are responsible for helping to maintain the structure of proteins and other biomacromolecules as well as the structure of cell membranes. However, before we begin to discuss these interactions it is worthwhile considering the properties of the solvent in which these interactions are usually most prevalent…water!
Properties of Water
Water is arguably the most important liquid on earth. All biological organisms have a high water content and need a relative abundance of it to survive.
Nanoscience also exploits the dielectric properties of water to help stabilise particles and surfaces in solution (via electrostatic colloidal interactions which we will cover later on in this course).
Although to us, water may seem like a typical liquid, this is far from the case. When compared to other simple low molecular weight liquids it has high melting and boiling points. It also has a high latent heat of vaporisation and surface energy when compared to other liquids. These properties all point to the fact that interactions other than simple dispersion interactions are present in water.
These properties are due to the special nature of the interactions between water molecules.
If we consider atoms such as oxygen (O), nitrogen (N), fluorine (F) and chlorine (Cl) we find that they are highly electronegative (they tend to pull electrons towards themselves).
This has the effect of making covalent bonds with these atoms quite polar. They are therefore permanent dipoles.
Bonds between these atoms and hydrogen tend to have a greater polarity. Interactions between dipoles on different molecules are called dipolar interactions. When these dipoles are formed by hydrogen atoms connected to electronegative atoms they are referred to as hydrogen bonds. Hydrogen bonding is present in water and is responsible for determining many of its properties.
Note: The presence of electronegative atoms will tend to create dipoles on a molecule. However, if the arrangements of the bonds are such that all the dipole moments cancel out there will be no net dipole moment on the molecule and it will be apolar. In the diagram above HCl, NH3 and CH3Cl have dipole moments that do not cancel out and are thus polar molecules. However, although both BF3 and CCl4 have polar bonds in them, the symmetry in the distribution of these bonds means that the net dipole moment is zero and they are apolar.
How strong are Hydrogen bonds?
The energies associated with individual H-bonds are typically
10-40 kJmol-1 (0.1 to 0.4 eV)
Comparing this with the energies associated with other bonds we have
Metallic and Covalent Bonds ~ 500 kJmol-1 (5 eV)
Van der Waals (dispersion forces) bonds ~ 1 kJmol-1 (10 meV)
They are harder to overcome than simple dispersion interactions and are relatively stable at room temperature. The temperature controls whether these bonds break apart. If the temperature drops below T=0°C thermal fluctuations struggle to overcome the strength of the bond. As the Temperature climbs to T=100°C thermal fluctuations have an increasing probability of overcoming the bond to enable the molecules to move apart.
The structure of ice
The location of hydrogen bonds on molecules infers greater specificity than simple dispersion interactions (i.e. dipoles will only interact electrostatically with dipoles and charges on other molecules), but it also introduces some directionality to the bonds. Dispersion forces on the other hand act between all molecules and tend to be indiscriminate in the directions in which they act (i.e. they have spherical symmetry).
The directional nature of hydrogen bonds in water causes water molecules to pack into an open tetrahedral structure when it forms ice. When water crystallises, the average number of nearest neighbour molecules decreases from ~5 in liquid to 4 in ice (resulting in a lower density) as the structure becomes more open.
This is unusual as crystalline solids are usually denser than the liquid because atoms/molecules become more closely spaced in the solid state.
The structure of water
At low (~ room or body) temperature, the polar nature of the interactions between water molecules causes liquid water to retain some of this tetrahedral structure.
The hydrogen bonds are still present, but the molecules are more mobile. The interactions between H-bonds are dipole-dipole interactions.
Summary of properties of H-Bonds
H-bonds are dipole-dipole interactions between specific functional groups. As such they have some directionality
They have energies of ~ 4-16 kBTroom (dispersion energies ~ 1 kBTroom )
This makes them ‘sticky’ and more stable than simple dispersion forces but they can also be disrupted by thermal motion of the molecules
Hydrogen bonding in nanoscience
The above properties are highly desirable when trying to form networks of molecules that will self-assemble into large ordered structures. Directionality and stickiness are clearly very important for helping to build a structure of any kind. Without stickiness, the structure would not hold together and in the absence of directionality we would have no control over how the molecules would stick together i.e. we would end up with a horrible, blobby, amorphous lump of molecules. However, an equally important property which makes hydrogen bonds particularly useful for creating self-assembling structures is the fact that they are relatively weak (compared to interactions such as covalent bonds). This means that they can be occasionally broken by thermal motion of the molecules. The thermal motion of the molecules allows bonds to break and reform in such a way that the system find its most stable structure and does not get trapped in other less energetically favourable states because bonds cannot be broken again.
Molecules that form many H-bonds with neighbours have higher dissociation energies and are thus more sticky than molecules which form only a few of these bonds. Varying the number of H-bonds on a molecule can therefore be used to increase or decrease the stability of the structures that they form; Moreover, if the locations of the H-Bonds are carefully designed into the molecules, the specificity of the interactions between them can be increased and greater control can be obtained over the structures that they will form.
If the molecules are designed carefully, the directionality of H-bonds can be used to form supramolecular aggregates that self-assemble in solution or on a surface. A nice example of this is the case of melamine and a molecule called PTCDI (see figure on the right). Each PTCDI molecule is capable of forming three specific hydrogen bonds with a melamine molecule. However, melamine can form a total of nine of these H-bonds with PTCDI, meaning that each melamine molecule can be connected to three PTCDI molecules. As result, when these molecules are mixed in the presence of a flat (noninteracting) surface, they tend to form large areas of a hexagonal lattice, where the vertices and edges of the hexagons are made of melamine and PTCDI molecules respectively.
Hydrogen bonding in nature
Biological molecules also exploit the directionality of H-bond formation to produce rigid, structural elements that give these large molecules a well-defined shape. For example, proteins use hydrogen bonds to create well-defined structural elements such as helical structures and folded chain structures called beta-sheets. DNA also uses the specificity of hydrogen bonding between complimentary base pairs to produce its characteristic double helix structure.
Hydrophobic interactions
The hydrophobic effect is a difficult subject as it logically fits with two sections of this course. Here we give an explanation of the effect but we will revisit the topic later (“Entropic forces”). If you find the discussion of entropy initially confusing this is understandable. However, a more detailed explanation of this aspect is included later.
When a nonpolar molecule comes into contact with polar liquids such as water it can only interact via dispersion forces (~0.4 kBTroom). These are much weaker than the hydrogen bonds that water prefers to form (~4-16 kBTroom).
As a result of these less favourable interactions, water molecules will tend to reorient themselves near non-polar molecules to form a more ordered structure which maximises H-bonding called a Clathrate cage (see image on right showing a methane molecule surrounded by water molecules).
This increase in the local ordering of the molecules causes a reduction in the entropy of the water surrounding the non-polar molecules. There is a large entropic energy penalty (equivalent to ~10-30 kJmol-1 or approx 4-12 kT at room temperature) associated with increasing the local order (decreasing disorder) of water molecules around a nonpolar molecule. Since this entropic penalty is proportional to the area of contact between the nonpolar molecule and the water minimising this area will be favourable. For this reason, nonpolar molecules will have a strong tendency to cluster together in water, to reduce the total area of the nonpolar/water interface. This is the physical origin of the hydrophobic effect.
Two droplets of a nonpolar liquid in water will experience an effective interaction due to the need to increase the entropy (decrease the order) of surrounding water molecules. They can satisfy this criteria by sticking together and reducing their total surface area.
Note: The hydrophobic effect is not the result of a direct interaction between nonpolar molecules. It occurs as a consequence of the need to maximise the entropy of surrounding water molecules
There is still no satisfactory theory to explain the details of the hydrophobic interaction. This is because the interaction between two surfaces involves many layers of intervening molecules.
Measurements show that the interaction energy, U, decays exponentially with the separation between surfaces
where 1 is the interfacial energy with water, S, is the interfacial area and o is the range of the interaction
o ~1-2nm
The hydrophobic force
The hydrophobic force between two flat surfaces with a separation D is given by
where 1 is the interfacial energy with water, S, is the interfacial area and o is the range of the interaction. o ~1-2 nm
Hydrophobic interactions in nature
The hydrophobic interaction is also used by protein molecules to ensure that the molecules fold up in a specific way. Parts of the protein that are required to be in the centre of the molecule are hydrophobic, while those that need to be on the outside of the molecule are hydrophilic. When these molecules are placed in water, they ‘fold’ up in such a way that the hydrophilic parts are in contact with water while the hydrophobic parts cluster together in the centre of the folded structure. When combined with the hydrogen bonded structures mentioned above, these interactions lead to the formation of highly specific arrangements of the different structural elements in the protein molecules. This is important because the arrangement of structures such as beta-sheets and alpha-helices control the biological function of these large molecules.
As we will see later, the hydrophobic force is also important in the formation of biological (cell) membranes and in the self-assembly of surfactant micelles. The latter structures have numerous applications in nanotechnology.
7. Surface vs Bulk
Cohesive energy of bulk atoms and surface atoms
In the bulk of a material, atoms and molecules are surrounded by NBulk nearest neighbours.
If each interaction with the central atom/molecule has an energy, -u, the total interaction energy becomes
UBulk = -NBulku
When we create a surface in material and expose it to vacuum each surface atom/molecule has fewer nearest neighbours
Nsurf < NBulk
The interaction energy per atom/molecule is then
Usurf = -Nsurfu
There is therefore an excess amount of energy
U = Usurf -UBulk = (NBulk - Nsurf ) u
associated with each surface atom/molecule. Since this is positive it is unfavourable.
Surface energy is the excess energy required per unit area to create a surface in a vacuum
The energy required to create two new surfaces in a material, each having area S is
where __is the surface energy of the material in Jm-2.
Calculating the surface energy of simple materials from inter-surface potentials
The surface energy of a material can also be thought of as the energy required to separate two surfaces from their inter-atomic/molecular distance, Do, and to remove them to infinity.
In lectures we showed that the potential energy due to dispersion interactions between two surfaces, U, is given by
where A is the Hamaker constant (J) and S is the area of the surfaces (m2)
This is also the energy required to separate the two surfaces from a separation Do to infinity. We would expect therefore that the surface energy should be equal to half this potential energy (as there are two surfaces) per unit area.
This gives:
Do calculations agree with experiments?
Work of Adhesion
The energy required per unit area to separate two surfaces of materials 1 and 2 in a third medium (medium 3) is called the work of adhesion (W).
It is the energy required to break the interface between two materials and form two new interfaces.
Interfacial energy is the excess energy per unit area required to create an interface between two different materials.
The interfacial energy between materials 1 and 2 is represented by 12
Surface and interfacial tension
Surface/interfacial tension is the force required per unit length to extend a surface/interface (measured in Nm-1). This is actually the same thing as the surface energy. Consider:
The energy (dU) required to increase the area of the liquid surface in the diagram by a small amount Ldx is
The magnitude of the force exerted on the barrier is therefore
So
So surface energy is also the force acting per unit length at a liquid interface (also called surface tension). Similarly, interfacial energy and interfacial tension are the same quantity
Why is a suspended droplet spherical?
A sphere has the minimum surface area for any given volume of fluid. Any shape other than a sphere will therefore result in an increased surface area dA and a consequent increase in the free energy dG = γ12dA. Alternatively one could think of this in terms of the surface tension which pulls inwards equally in all directions leading to a spherical shape.
Wetting interactions and contact angles
If a droplet is placed on a surface the shape is more complicated. Surface and interfacial energies determine how macroscopic liquid droplets deform when they adhere to a surface
Partially wetting films (S < 0)
S = γvs – γLs - γvL
Wetting film (S > 0)
The contact angle made between a droplet and a surface is determined by a subtle balance of the interfacial energies/tensions.
Cohesive v Adhesive Forces
The angle a liquid makes with a surface is controlled by a balance between the cohesive and adhesive forces.
“Cohesive” – interactions with other molecules of the same type
“Adhesive” – interactions with other molecules of a different material or liquid
If a surface has a small / large contact angle with water then it is known as “hydrophilic” / “hydrophobic”. Small and large are relative terms used to compare different surfaces but usually a hydrophobic surface would have a contact angle ~ 90˚.
Superhydrophobicity
Under certain situations if a surface contains microstructures it is possible to achieve contact angles as high or greater than 160˚. This is due to the fact that only a small portion of the droplet actually touches the surface. Under these circumstances it is possible for the droplet to simply roll off the surface with even a small angle of inclination. These structures occur naturally (eg the Lotus Leaf) but are also especially manufactured for various applications.
Voids filled with air
Summary of key concepts
Surface/Interfacial Energy is the energy per unit area associated with the creation of a surface/interface (Jm-2)
Surface tension (Nm-1) = Surface energy (Jm-2)
Surface energies can be calculated using the latent heat of vaporisation of a material or from the inter-atomic/molecular potentials
A subtle balance of interfacial energies determines the adhesive properties of simple materials and the wetting properties of a liquid.
8. Capillary Forces
Pressure difference across a liquid interface
What happens if the surface between two liquids becomes curved? Consider, the water-air interface of some water in a tube.
In the last lecture we showed that there is an energy cost associated with increasing the area of a surface. A curved surface has a larger area than a flat one.
To study this question in more detail, we will use the principal of virtual work. That is we will ask the question, how much work would we need to do to change the curvature of the liquid interface? This work must then be equal in magnitude to the potential energy stored in the curved interface. From this information we will be able to determine the magnitude of the forces or the pressure (force per unit area) acting at the interface.
We start by considering a curved liquid/air interface which has two principal radii of curvature, R1 and R2, (see diagram on right).
There is a pressure difference across a curved liquid interface which acts to try to reduce the area of the interface. This is called the Capillary Pressure
where is surface energy/tension (Jm-2)
R1 and R2 are principal radii of curvature in two orthogonal (perpendicular) directions
Capillary Rise
When a fine capillary is placed inside a liquid, a curved liquid meniscus forms. The resulting pressure drop across the interface causes the fluid to be drawn up inside the capillary. This is phenomena is referred to as capillary rise and can be observed routinely in a laboratory.
The height of fluid (h) in a capillary of radius, R, is determined by balancing the capillary pressure drop across the meniscus with the pressure required to draw the fluid up the capillary.
For a fluid of density,, and surface tension,
Summary of key concepts
Capillary pressure due to a curved liquid interface
Capillary pressure is responsible for the phenomenon of capillary rise
9. Nanoscale Force Measurement Techniques – AFM
Forces at the Nanoscale
Forces between nanoscale objects typically occur in the range 1pN-1nN and can be difficult to measure.
To measure these forces we ideally require a measurement system where the change in applied force gives a linear response in the measurement device (although this is not necessary).
In this chapter we will study experimental techniques that can be used to measure nanoscale forces directly. The first of these techniques is Atomic Force Microscopy
Atomic Force Microscopy (AFM)
In the first year ‘Frontiers in Physics’ module, you considered how atomic force microscopy can be used to image the surface of a sample. There are two main imaging modes in AFM:
Contact mode
A sharp tip is mounted on a cantilever and brought into contact with the surface to be studied. As the tip is dragged along the surface, the cantilever deflects. If we track the position of the cantilever we can build up an image of surface as the tip moves along. The contact between tip and sample can be quite destructive, so this mode of imaging is usually reserved for hard samples.
Tapping mode
A tip mounted on a cantilever is oscillated at its resonant frequency and brought close to a surface. The tip-sample interaction causes the resonant frequency of the cantilever to shift by an amount which depends upon the tip-sample separation and the local gradient of the force. If we move the tip across the surface and vary the tip-sample separation in such a way that the resonant frequency remains constant, we can build up images of constant force gradient. For a uniform, chemically homogeneous sample this corresponds to an image of the topography (or height variations). We can thus build up images of the surface structure in a similar way to that used in contact mode. However, the advantage of this technique is that the tip spends very little time in contact with the surface and is less destructive. This technique is therefore more suitable for imaging soft samples.
Force Measurements using the AFM
AFM can also be used to measure forces. If an AFM tip is brought into contact with a surface, the forces on the tip will cause the cantilever on which it is mounted to deflect.
The AFM cantilever bends/deflects in response to an applied force. The applied force is given by
The spring constant of the cantilever can be derived using the physics of elastic beams and is given by
where E is the Youngs Modulus of the cantilever (Pa), l is its length and I is its geometric moment of inertia (m4).
Detecting the deflection of the cantilever
If small (pN - nN) forces are exerted on the cantilever, then the displacements of the cantilever are likely to be small (pm - nm). Our ability to detect small forces is controlled partly by the design of the cantilever and partly by the instrumentation used to detect the deflections of the cantilever.
The most common method of detecting cantilever deflections is to reflect a laser beam off the back of the cantilever.
The laser light is collected using a position sensitive photodetector, called a split photodiode, which produces an output current that varies with the position of the laser beam on the detector. These detectors are simple two photodiodes (PD) that are connected to a differential amplifier in such a way that the output signal is proportional to the difference in intensity detected by the two photodiodes. The diagram below shows a laser spot falling on a split photodiode detector and a bar graph showing the measured intensities at the two photodiodes.
Photodiode
current
The advantage of using this type of detection system are
these detectors are relatively inexpensive
the linear deflection of the laser beam on the split photodiode depends upon geometry (i.e. the length of the cantilever and the cantilever-detector distance). This means that the detection system can be optimized to give the best response for a given force range by changing these lengths/distances.
Interpreting the photodiode signal
As mentioned above, the split photodiodes are connected to a differential current amplifier which produces a signal that is proportional to the difference in photocurrents generated by the two diodes
If isignal > 0 force acts upwards
If isignal < 0 force acts downwards
The detection efficiency of the photodiodes is such that cantilever displacements as small as 0.1 nm can be routinely detected. This means that forces in the sub nN range can easily be measured using AFM (see problem 6.1 above).
Calibrating the spring constant of the cantilever
In practice, it is often difficult to calculate the spring constant of a cantilever, so it has to be calibrated prior to a measurement. Reasons for this might include the fact that the Young’s Modulus or dimensions of the cantilever are not known with sufficient precision to calculate the spring constant.
Calibration: Thermal tuning of the cantilever
This method involves measuring the natural response of the cantilever in response to thermal motion. When the cantilever is agitated by thermal motions of the atoms around it, it starts to vibrate. Vibrations at the resonant frequency of the cantilever occur much more easily than vibrations at other frequencies (which are heavily damped).
If we monitor the displacement of the cantilever z obtained from the noise signal when the cantilever is not being driven, we obtain something like the signal shown below
z
We can calculate the mean square displacement of the cantilever <z2> from the data shown in the plot. We then assume that the potential energy stored in the cantilever is comparable to the thermal energy provided by its surroundings to give
Note: the factor of ½ on the right hand side comes from equipartition theory (more on this in Thermal Physics), but the thermal energy is still of order kBT, where kB is Boltzmann’s constant. Rearranging this equation gives
Hence the spring constant of the cantilever can be determined using a knowledge of the temperature, T, and measurements of the mean square displacement of the cantilever under the influence of natural thermal vibrations.
Force detection limits in AFM
Typical cantilever spring constant: 0.01-50 Nm-1
We can routinely measure cantilever displacements of ~ 0.1nm
We can therefore measure forces > 10 pN with relative ease using AFM
Force vs. distance curves
AFM can be used to map out the strength of dispersion (and other) forces as a function of the distance between sample and tip.
Protein (un)folding
AFM is routinely used to map out the forces associated with the folding and unfolding of individual protein molecules
Moving the AFM tip with nanoscale precision – Piezo electric motors
In order to build up images with an AFM one must scan the tip backwards and forwards across a sample building up a 2D map of the force measured at different points (see figure).
Diagram illustrating how an AFM builds up an image of a sample
In order to do this successfully one must be able to move the position of the tip with sub-nanoscale precision. AFMs and other similar technology accomplish this using motors based on the piezo-electric effect.
Certain kinds of crystal are called piezo-electric crystals (e.g quartz). These crystals become polarized when subjected to a stress / strain.
-
-
-
-
-
-
-
-
E
Force
The diagram above shows how when a crystal containing equal numbers of positive and negative elements is deformed by an external force it results in the formation of a dipole. As a result of the dipole an electric field is developed across the sample.
The same effect however can work in reverse. If an electric field is applied across the sample this will deform the sample due to the forces acting on +ve and -ve elements resulting in expansion or contraction along one dimension.
-
-
-
-
v
-
-
-
-
Force
E
This controlled expansion is usually small. By stacking these crystals and electrodes it is possible to make very precise motors capable of controlled motions well below a nanometre by changing the applied voltage.
The changes in a piezo-electric crystal can be related by the following equations.
P is the polarization, d is the piezo-electric constant, α is the polarizability of the material, E is the electric field. ɛ is the linear strain ie change in length / original length. Y is the young’s modulus.
10. Nanoscale Force Measurement Techniques – Optical Tweezers
Optical Tweezers
Optical tweezers provide a way of manipulating atoms and molecules, as well as nano or micron scale objects. They can also be used to measure very small forces.
These instruments work by trapping small particles in a focussed laser beam. Once trapped, particles can be manipulated by simply steering the laser around.
The wave nature of light
Light is an electromagnetic wave. As it propagates through vacuum or a medium it carries energy. The energy is stored in electric and magnetic fields that oscillate in a direction perpendicular to the direction of travel of the waves (more on this in your ‘Classical Fields’ module).
As the light travels along, the Electric field at a given time, t, and point, x, in space will vary according to the equation
where Eo is the amplitude of the wave, ω is its angular frequency and k is called the wavenumber (_=2π/λ_where _λ_is the wavelength of the light).
Intensity
The time averaged intensity of light, I, can be related to the magnitude of the electric field, E, by the relationship
in Wm-2
Where c is the speed of light (ms-1) and n is the refractive index of the medium in which it propagates. All other constants have their usual definitions.
Note: For a sinusoidal oscillating field, the time average value of <E2> is not zero. So the intensity is not zero! In fact, it is equal to ½ E02. The proof of this is left as an exercise for the reader.
The key point here is that intensity is proportional to the electric field squared
Optical trapping of particles
Small dielectric (insulating) particles experience a force when illuminated with visible light. If a dielectric particle is placed in an intensity gradient a force is exerted on the particle such that it will move to a region of higher intensity.
To understand the origin of this effect we must consider the interaction between the dielectric particles and the incident laser light radiation.
When a particle is placed in a laser beam, the electric field associated with the incident light will cause a slight separation of charges in the particle material and thus induce a dipole moment on the particle. This dipole moment will then interact with the field which caused it to form.
So in one dimension the force on a dielectric particle is
Generalising this to three dimensions we obtain
So the force always acts along the direction of increasing intensity gradient. Particles will therefore tend to move to regions of higher intensity.
Making a simple optical trap
We saw in the last section that dielectric particles will tend to move to regions of higher intensity in a laser beam. How can we use this to create a simple optical trap?
The answer to this question is extremely simple. If we consider the intensity profile inside a laser beam, we find that it is not uniform across the beam, but instead it has a Gaussian shape as we move from the centre of the beam to the edges.
where Io is the intensity at the centre of the beam, wo is a measure of the beam width and r is the distance from the centre of the beam.
This Gaussian profile means that an intensity gradient already exists within the laser beam and small dielectric particles will tend to be attracted and trapped in the centre of the beam.
However, the problem of trapping along the axis of the laser beam (z direction) still needs to be considered as no intensity gradient exists for a coherent parallel beam. This difficulty can be overcome by focussing the laser beam using a simple microscope objective lens.
When the light is focused, it creates a trap along the axis of the beam. This is because the area of the beam decreases as it approaches the focal plane and then increaeses after passing through it. This causes the intensity to steadily increase and then decrease on passing through the focal point of the lens.
The strength of the trap in the z direction can be varied by changing the angle, , (see diagram on left). The larger this angle, the steeper the intensity gradient will be along the axis of the laser beam. The maximum gradient in intensity can be obtained by using a lens with a high numerical aperture (NA), where NA is defined as
where n is the refractive index of the ambient environment.
Trap stiffness (Quadratic approximation to beam profile)
For small displacements from the central position we can approximate the Gaussian intensity profile of the beam using a quadratic function
Laser Intensity
Intensity gradient
As we will see in the following examples, this allows us to determine the effective spring constant (or stiffness) of the trap and to use optical tweezers to measure forces that are exerted on trapped particles.
For small displacements (r<<wo) the intensity profile can be approximated by a quadratic form. This results in a force which is directly proportional to the displacement of the particle within the trap.
Where
Hence, if we know the displacement and the trap stiffness, we can calculate the force exerted on our trapped particle.
Detection of particle movement
The split photodiode arrangement used in AFM can also be used to detect deflections in optical traps. However, a quadrant photodiode is used to detect deflections in both the x and y directions.
In the case of large particles, a shadow is cast on the quadrant photodiode. This means that measured intensity varies as the particle moves in the x-y plane.
For smaller particles, light scattered from the particle interferes with light in the main beam to produce a fringe pattern on the photodiodes. As the particle moves around, the fringe patterns move. This can also be used to detect motion of the particle.
Quadrant photodiode
(4 photodiodes)
Shadowing (large particles)
Fringe pattern (small particles)
Laser beam
Calibration of optical traps
There are 2 main techniques used to calibrate the spring constant or ‘stiffness’ of optical traps.
Method 1) We can measure the displacement of a particle under the influence of known forces
Method 2) We can measure the mean square displacement, <x2>, of a trapped particle under the influence of thermal motion
Force Measurements
It is possible to perform force measurements on e.g. single molecules.
An optical trap and a fine micropipette can be used to measure the forces exerted by individual DNA molecules.
Small polymer beads can be tethered to the ends of the molecule and used to manipulate it.
C. Bustamante et al., Nature, 421(23), 423 (2003)
Summary of key concepts
Atomic force microscopes can be used to measure forces,
The spring constant of an AFM cantilever is determined by its material properties and its physical dimensions
A split photodiode arrangement is used to detect deflections and to measure forces with >10pN precision
Particles in a laser beam will experience a force that acts to pull them to regions of higher intensity
Lasers have a Gaussian beam profile, which naturally lends itself to trapping of particles.
Trapping in three dimensions can be achieved by focussing the laser using a microscope objective.
Trapping forces on nanoscale particles can be measured with pN precision.
10. Colloidal Stability & Osmotic Pressure
Colloids and nanoparticles
Colloids are mixtures where small particles (<1 m in diameter) of one substance are suspended in a second medium or continuous phase. They can be made of solid particles suspended in a liquid or liquid droplets suspended in another liquid (also called emulsions).
They are not solutions! The material is not completely dispersed, but exists as discrete particles.
Colloidal interactions govern the properties of nanoscale particles and interactions between surfaces in liquid environments. Understanding and controlling these interactions is therefore essential in the manufacture of nanoparticles, ferrofluids (magnetic nanoparticles) and solution based quantum dots. These interactions are also important in mediating the interactions between large biological molecules such as proteins and play important role in stabilising the nano to micron scale fat droplets in milk.
The key challenge in manufacturing nanoparticles and colloidal dispersions is preventing the interparticle/intersurface forces from causing the small particles from sticking together. Moreover, these systems have a very high surface to volume ratio when compared to bulk materials. This means that there is a large energy penalty associated with the large surface area of these dispersed systems.
The natural tendency is for small particles to stick together or flocculate to form an aggregate (more on aggregation in a later lecture) and hence reduce their total surface area. We have already seen that dispersion forces result in a mutual attraction between particles. These can drive the flocculation of even very dilute systems.
How do we stop things sticking together?
We need to introduce a short-range repulsion force to keep particles apart and prevent them flocculating under the action of dispersion forces. In water, this is often done by decorating the surface of the particles with charged groups and adding a small amount of salt.
When the particles are added to salt water (or electrolyte) the dissociated salt ions cluster around the charged particles to form a counterion cloud. The counterion cloud can be thought of as having two regions. The first is the Stern (also sometimes referred to as the Helmholtz) layer. This layer consists of ions that carry the opposite charge to the surface groups on the particle surfaces and is usually tightly bound to the surface. The second layer is called the diffuse double layer. This layer consists of a mixture of positive and negative counterions which are relatively mobile and are distributed in such a way that the system becomes charge neutral.
If the particles are charge neutral then there can be no net electrostatic interaction between them. So how does decorating the surface of particles with charged groups introduce a repulsive force between them?
The answer to this question is that the presence of the counterions around the particles creates an osmotic pressure between the particle surfaces.
Pure water
Water with particles
Semi-permeable membrane
Flow of water
Osmotic Pressure – the pressure arising from spatial gradients in the concentration of a solute. This pressure acts to restore a uniform concentration and will cause a flow of water unless or until a sufficient pressure acts to oppose it.
When the particles/surfaces approach the local concentration of counterions between them increases relative to the bulk concentration of counterions (i.e. far away from the particles).
If we think of the charge neutral system of ions floating around in between the two particle surfaces as a non interacting (ideal) gas of ions we can use the ideal gas law to derive an expression for the pressure exerted by the counterion clouds. However, before we proceed it is worth mentioning that although the electrostatic potential energy of the charged particles does not influence their interactions directly in an electrolyte, it does determine how the counterions will be distributed around each particle. As we will show below, this has some influence on the local concentration of counterions and hence the osmotic pressure exerted between the particle surfaces.
The osmotic pressure, , which pushes the surfaces apart and tries to restore a uniform counterion concentration is given by
where n+ and n- are number densities of positive and negative ions in the gap, and no is their concentration far away from the surfaces. Note that for an electrolyte (salt) in bulk solution (e.g. NaCl) the concentration of positive counterions (Na+) is equal to that of the negative counterions (Cl-).
How do we calculate n+ and n-?
We have deliberately assumed that the number densities/concentrations of positive and negative counterions in the gap between the two surfaces will be different. This is not unreasonable if we consider our diagram of the stern and diffuse double layers above, as we might expect an imbalance in these concentrations to occur for charge neutrality to be maintained.
To determine the number densities of the counterions we need to consider the electrostatic potential energy of the counterions in solution and how this influences their local concentration.
For an ion of charge ± ze, sitting in a spatially varying electrostatic potential V(x)the energy, E, is simply given by
To determine how this energy influences the distribution of counterions, we recall that the probability, p, of a particle occupying a given energy state can be described by Boltzmann statistics (See thermal physics notes)
In this case, the energy of the particle is determined by its position in the potential V(x) (as described above) and the local number density must scale with the probability associated with finding the particle with a given energy. Hence we can simply write the number densities of positive and negative counterions as
Hence, our expression for the osmotic pressure becomes
In the limit of small surface potentials i.e. when this reduces to
So if we know how V(x) depends on the distance away from the surface we can calculate the osmotic pressure due to the counterions.
What form of V(x) should we use?
The true form of the electrical potential between colloids and charged surfaces is complicated. However, a good starting point is to approximate V(x) to the form expected for the potential due to an isolated charged surface in an electrolyte.
To obtain this we must first solve the Poisson equation which relates the electrostatic potential V(x) to the local charge density, , using the following formula.
where and o, are the relative permittivity of the medium in which the counterions are dissolved (usually water) and permittivity of free space (Fm-1) respectively.
Potential near a planar/flat charged surface
The charge density associated with the counterions is simply the product of their charge and number densities;
In the limit of small potentials, , the Poisson equation for the cloud of counterions becomes
Solution of the Poisson equation and the Debye screening length
The Poisson equation for the counterion cloud is very similar to the equation for simple harmonic motion (a=-2x) and therefore must have a solution which consists of a sum of positive and negative exponentials. However, we require that the potential decays to zero for infinite, x. (If it didn’t, the total energy of the counterion cloud would be infinite!). This means that we are left with a solution which consists of a simple negative exponential of the form
Where Vo is the surface potential (i.e. the electrostatic potential at the surface) and is a decay constant such that if we insert V(x) into the Poisson equation we obtain
1/κ is the Debye screening length. (N.B the Debye length is 1/κ not κ!)
It is the distance over which electrostatic interactions are screened in an electrolyte and Vo is the potential at the surface.
Osmotic pressure between surfaces (revisited)
Recall that the osmotic pressure due to counterions between charged surface has the form
where
If we insert our solution for the potential we obtain
where D=2x is the distance between the two surfaces.
This expression describes the positive (repulsive) osmotic pressure caused by the condensation of counterions in the gap between two surfaces.
Total pressure between surfaces
The total pressure between two charged surfaces in an electrolyte is given by summing the contributions due to repulsive osmotic pressure and attractive dispersion forces to give
where Pdispersion is the pressure due to the attractive dispersion interactions. We have
where the first term is due to the osmotic pressure and the second due to the attractive dispersion forces.
This is a simplified form of the equations derived by Derjaguin, Landau, Verwey, Overbeek (DLVO) when formulating the DLVO theory of colloidal stability.
The plot below shows how the dispersion pressure, the osmotic (double layer/counterion) pressure and total pressure vary with the separation of the surfaces.
At short range (<10 nm) the repulsive pressures are high until below a critical separation of ~4 nm the pressure becomes attractive as dispersion forces dominate. It is the large positive pressures in the range 4-10 nm which stop the surfaces from sticking together when they approach. However, if the surfaces are pushed together with pressures that exceed ~4000 Pa, the surfaces will adhere to one another.
The inset shows that at long range (>18 nm in this case) the pressure is weakly attractive and that at a separation of ~18 nm there is no net pressure on the surfaces. This corresponds to the equilibrium separation of the surfaces under these conditions.
The effects of adding electrolyte
As more salt is added, electrostatic effects are more strongly screened → eventually attractive dispersion forces dominate and surfaces will stick together. This is manifested as a reduction in the height of the repulsive pressure ‘barrier’ until finally, the pressure becomes negative for all values of the inter-surface separation, D.
Increasing salt concentration
Summary of key concepts
Colloidal stability is important in many areas of nanotechnology
It can be achieved by charging surfaces
The pressure exerted on charged surfaces in an electrolyte is determined by a balance between the osmotic pressure due to counter-ions and the dispersion pressure .
11. Entropic Forces
Colloidal Stability
In the last lecture we discussed how colloidal particles can be stabilized against flocculation by weakly charging the surface of the particles. However, it is sometimes not possible to use charges to keep particles and surfaces apart in solution. This is particularly true when using organic solvents as the suspending medium (e.g. in ferrofluids), as ionic solids (electrolytes) are often poorly soluble in these solvents and thus unable to stabilise the system.
An alternative route to stabilisation is to use Steric (or entropic) repulsion effects between surfaces. This stabilisation method involves decorating the surface with long molecules (e.g. polymers). The molecules are tethered to a surface by one end and can wave around freely in solution. As a result they have a large number of possible orientations/configurations (i.e. directions in which they can point).
When a second surface comes close to the surface that is covered with our molecules, the number of configurations that the molecules can adopt is reduced. This reduces the entropy (or measure of disorder) of the molecules. Recall from your ‘Thermal physics’ module that the second law of thermodynamics states that the ‘the entropy of an isolated system can never decrease’. This means that there is a tendency for the entropy to increase and to be maximised in an isolated system. (We will assume for the sake of argument that the tethered molecules represent an isolated system).
If the entropy of the tethered molecules is reduced by bringing a second surface into close proximity, the system will tend to resist this reduction in entropy by generating a repulsive force. This entropic repulsion is the physical origin of steric forces between surfaces.
Boltzmann form of Entropy
Boltzmann derived a statistical form of the entropy of a system, S, such that:
where W is the number of available (micro) states corresponding to a given macrostate and kB is Boltzmann’s constant (JK-1 ).
In this case…
Macrostate → molecule tethered to the surface
Microstate → a single orientation of the molecule
Calculation: the entropy of a tethered rod
Suppose we treat the molecule as a rigid rod with length, l, and cross sectional area, a. If we allow the rod to point in any direction it can sweep out half a sphere with a total area, A1
Consider the free rod (ie when no upper surface is present)
Calculate the Area of the surface it can move over
Work out the number of microstates W1=A1/a
Use the Boltzmann formula to calculate the entropy S1
Repeat the calculation for the rod with upper surface at D < l
Calculate the Area of the surface A2 it can move over
Work out an elemental area in terms of xSubstitute for x in tems of l and cosθ
Integrate between θmin and θmax
Work the number of microstates W2=A2/a
Use the Boltzmann formula to calculate the entropy S2
The important quantity is the change in entropy with separation D of the two surfaces
ΔS = S2 – S1
For the case considered in the lecture where θmin = 0 and θmax is defined by the distance of the upper plate
Entropic repulsion force
We can extract a force if we know how the entropic contribution to the interaction energy , U, depends upon separation D. As
To determine the energy change associated with our change in entropy we must determine the change in free energy associated with bringing the two surfaces together. The change in the Gibbs free energy, U, (see ‘Thermal physics’ notes) of the system at constant temperature is given by:
The first term, H is the change in enthalpy. Essentially this encompasses all the interactions we have met thus far in this course such as the formation of bonds.
The second term TΔS is the change in energy due to entropy. A decrease in entropy results in an increase in the free energy U. Therefore, if there are no changes in H this will result in a repulsive force.
How does entropy result in a force?
Whilst it is clear mathematically that entropy may result in a force, it is helpful to understand this fundamental idea on a conceptual level. It is often stated that entropy is a measure of disorder. This statement whilst true needs clarifying. Entropy is actually a statement about the number of possible equivalent configurations.
Consider the following analogy: a bedroom can be organised in many different ways. Clothes could be strewn across the floor or folded on a chair, the lamp could be in any number of different positions, the books could be on the shelf or chucked on the floor. All of these different arrangements would represent a single configuration or microstate. However, there are many more ways in which a bedroom could be untidy rather than tidy. In making this statement we are saying that the different ways in which a room is messy are functionally equivalent. It doesn’t matter which way, we would still call it untidy. In terms of Boltzmann’s entropy formula we might say that an untidy bedroom has a larger entropy than a tidy one. This is because the number of equivalent configurations or microstates of an untidy room is larger than a tidy room.
However, here is a key point. How does a tidy bedroom go from tidy to untidy? The fact that there are more messy room configurations than tidy ones by itself does not explain why a room might end up messy. In our equation above the free energy depended on ΔS ( a statement about the change in the number of configurations) but it also depends on Temperature. The role of temperature is to allow the system to freely explore all the different configurations. In the case of our room we might say that temperature is analogous to a strong earthquake. The earthquake enables all the items in a room to move around, creating (or “sampling”) different configurations. Now the role of entropy is clearer. Since the earthquake causes the configurations to be sampled randomly the final configuration is a matter of chance or probability. Since there are many more messy configurations we expect that the room will become messy if hit by an earthquake. The explanation depends both on entropy ( a statement about probability) and temperature (a way of sampling those different microstates).
Finally, consider the situation in which you are in a tidy room when an earthquake hits. In order to keep the room tidy (to work against the change in entropy) requires you to do a lot of work, to apply forces to prevent things moving out of place. You need to apply forces to resist the “entropic force” being generated.
In the microscopic world of colloidal particles, salt ions and nano-rods tethered to surfaces interactions are comparable to the thermal energy kBT. This means the temperature enables particles etc to sample different microstates giving rise to significant entropic forces.
The pressure between two surfaces
Earlier we calculated the entropic force produced by a single rod tethered to a surface. In reality a surface will be covered with a large number of rods with an average spacing.
In the case where the separation between tethering points is, d, then the effective area occupied by one molecule is d2
The force per unit area (Pressure) is P, where
Summary of key concepts
When the motion of molecules near a surface is confined it reduces the entropy of the system.
This gives rise to a repulsive force which acts to push surfaces apart and oppose the entropy reduction
For rod like molecules on a surface which occupy an area d2, the pressure between surfaces at a separation, D, is given by
Depletion Forces
Real systems can often contain multiple components of different sizes, think for example of a biological cell with many different sized proteins or a paint where there are both small pigment particles and large latex particles. Even in the absence of specific energetic interactions between these components the large particles / surfaces can experience an attractive force due to the entropy of the smaller particles.
The motion of a large particle is filmed as it diffuses inside a boundary. In (b) only the large particle is inside the boundary. The probability of finding the particle at a particular location is indicated by the colour map. The probability is uniform across the entire volume. In (c) lots of small particles (which are not visible) are also added to the volume. Now the large particle spends nearly all of its time trapped against the boundary. This indicates that there is now a force keeping the large particle at the edge. [ Dinsmore et al PRL 1998]
Excluded Volume
r
r
In a multi-component system, the motion of the small particles in a suspension is restricted by the presence of the larger object (eg a large particle or plate). In addition to the volume taken up by the large object itself the small particle centres are also “excluded” from a thin layer around the object related to their own size. The entire volume (volume of the large object and the layer around it) is known as the excluded volume.
Key insight: Small particles are excluded / depleted from a volume around any larger object in solution.
Depletion Force between 2 plates
Two large objects in a suspension of smaller particles can experience an attractive force if they come too close together. This force arises from the change in entropy of the small particles arising from the change in excluded volume around the larger objects as they come close together.
Calculation: depletion force for 2 flat plates in suspension of small spheres
Calculate the excluded volume of the plates when they are far apart (Vexcl_1). This is related to the geometry of the object and the geometry of the small particle (see above)
Calculate the excluded volume when the objects come close enough together that there excluded volumes overlap (Vexcl_2).
Calculate the change in excluded volume. ΔVexcl=Vexcl_2 - Vexcl_1.
Be careful with the sign of this term, it should be negative as the objects come closer together since the overlap region between the objects reduces the total volume.
The change in volume accessible to the particles ΔVp increases as the excluded volume decreases. There are therefore 2 scenarios:
D > 2r à No change in Vexcl since no overlap
D < 2r à
This increase in the volume accessible to the small particles, means there are more possible configurations or microstates the closer the large plates come together.
From our previous lecture we know that S=kBlnW à ΔS > 0 as the separation between the plates, D, decreases.
Consequently:
This implies that if the plates are separated by a distance D < 2r there will be an attractive force between the plates.
Calculating the number of microstates is however tricky. We showed in lectures (this will not be examined) that this problem can also be equivalently described in terms of an osmotic pressure associated with the small colloidal particles. This lead to a key result:
Entropic Force v Osmotic Pressure
The Depletion force can therefore be described in 2 different but equivalent ways:
It arises due to the increasing entropy of the small spheres due to the increased volume as the plates move together.
It is an osmotic pressure difference acting on the plates due to the different concentration of small particles in between and outside the plates.
It is perhaps easy to get confused about whether the Osmotic pressure results in attraction or repulsion. We have seen that for 2 charged plates the pressure is repulsive, but for 2 plates surrounded by small spheres it is attractive. The pressure acts to reduce the differences in concentration. Thus in the charged plates the concentration of ions is higher between the plates. The pressure therefore draws liquid from the bulk into the gap, pushing the plates apart and reducing the ionic concentration there. In this lecture the concentration of particles is high outside the gap and zero in between. The pressure therefore pulls water from the gap into the bulk to reduce the concentration. This in turn pulls the plates together to compensate for the loss of water in the gap.
Depletion Force 2 spheres
The depletion force between 2 large spheres in a fluid of smaller spheres can be calculated in an analogous way to the 2 plates. The problem is the same, since we must work out the change in excluded volume between the two spheres as they come close together. However, the different geometry makes the calculation a little trickier.
R
D
The volume of overlap between 2 spheres is twice the volume of a spherical cap.
R’
h
h = R’ – D/2 and R’ = R + r
Hydrophobic Effect revisited
In section 6 it was highlighted that the hydrophobic effect was a difficult subject since it fitted with different sections of the course. Hydrogen bonds are permanent dipoles and hence we initially visited the subject in section 4. However, in understanding the hydrophobic effect we also needed to consider a principle which we have subsequently explored, that is that Entropy can produce a force (section 11)
Now hopefully with a broader understanding of this principle we can revisit the hydrophobic effect and show how this fits together.
Let us consider a single water molecule surrounded by other water molecules. The single water molecule is composed of Oxygen and Hydrogen atoms resulting in a permanent dipole. This permanent dipole forms because the electronegative oxygen atom pulls the electron density in the covalent bond towards itself. This results in a slight positive charge on the Hydrogen and slight negative charge on the Oxygen. These charges interact with neighbouring water molecules so that there is a “Hydrogen Bond” between the Hydrogen of one molecule and the Oxygen of a neighbour.
The Hydrogen bonds are much more stable than dispersion interactions but still able to be disrupted by thermal fluctuations on a reasonable timescale. Typical Hydrogen bond energies ~ 12kBT when compared with dispersion interactions ~ 2kBT. This means that in liquid water the water molecules find a stable orientation for a small amount of time and then change orientation or neighbours. Now let us think of this in terms of the language of entropy we have been developing. Since there are lots of orientations and lots of possible neighbours this means there are a large number of microstates for a collection of water molecules.
Now consider adding a single non-polar molecule to the collection of water molecules. Water molecules cannot form hydrogen bond interactions with this molecule and so to minimise the number of lost hydrogen bonds the water molecules must arrange themselves in a very specific structure – “The clathrate cage”. How many equivalent configurations are now accessible to these water molecules? Due to the restrictions imposed by the presence of this non-polar molecule the water molecules cannot be arranged in lots of different orientations and they are confined to a relatively small number of nearest neighbours. In other words the number of microstates is much smaller than before.
If we add a second non-polar molecule to the water this likewise results in a load of new water molecules reducing their entropy. We should be able to see that for each bit of surface area of each non-polar molecule a number of water molecules have a reduced number of configurations. If the two non-polar molecules come close enough together the number of water molecules affected starts to reduce. So we therefore have some water molecules whose orientations and mobility is no longer restricted increasing their entropy. This is approximately proportional to the accessible surface area of the two non-polar molecules.
Summarising: when the two non-polar molecules are separate:
large exposed surface area
restricts number of configurations of large number of water molecules
small number of microstates
small entropy
As they come together:
reduced exposed surface area
less water molecules have restricted configurations
slightly larger number of microstates
increased entropy
Earlier we derived the expression:
which relates the force to the change in entropy with distance. In this case as we reduce the distance between the non-polar molecules the entropy of the system of water molecules surrounding them increases. If ΔS > 0 and ΔD < 0 then F < 0. Ie the change in entropy of the water molecule results in an effective attractive force between the two non-polar molecules.
An additional clarification which will not be examined:
Our picture of the clathrate cage is somewhat simplistic. It implies that only the layer of water molecules in direct contact with the non-polar molecule is affected and hence contributes to the change in entropy. This is not true. A number of layers of water molecules will have a restricted set of configurations. To understand why consider how the second layer of water molecules interacts with the first layer. Since the first layer cannot easily rotate, the second layer must form a certain relative angle to be able to form hydrogen bonds. This effect gradually reduces as one moves further and further away due to fluctuations. This is why there is still a hydrophobic effect between two plates which are more than two water molecules apart. As the plates move closer together the number of molecules affected reduces (the entropy of the entire system goes up) since the available volume and hence number of water molecules in the gap also decreases.
12. Aggregation & Self-assembly
Aggregation We have seen that particles and surfaces can interact via many different physical interactions (e.g. dispersion, double layer and steric forces). The subtle balance between these interactions determines whether the particles/surfaces will remain separated in solution.
When the interactions between particles favour the attraction of particles, they will tend to aggregate. If these interactions become really strong, they will eventually drive phase separation in the system and the components will unmix forming effectively infinite aggregates.
Why do aggregates form?
When the energy per particle/molecule associated with a particle/molecule inside an aggregate is less than the energy of a free molecule, then aggregates will start to form. If this average energy becomes lower (more favourable/negative) the bigger the aggregate gets, then the aggregate will continue to grow.
The shape of the aggregate that forms will depend upon the nature of the interactions between individual molecules. If for example, the molecules have highly directional interactions at opposite ends, they will tend to link up to form long chains or one dimensional aggregates. If the molecules interact in two dimensions they will tend to form disks or planar aggregates. And if the interactions are isotropic (i.e. the same in all directions), the molecules will tend to form three dimensional aggregates, droplets or clusters.
Chemical potential of a molecule
In order for an aggregate to grow the change in the ‘free’ energy when an additional molecule or particle is added must be negative. We define the chemical potential μ of an aggregate as the rate of change of the free energy with particle number N.
ie
As we will see the chemical potential depends upon the shape of the aggregates and the number of particles in the aggregate (N)
One Dimensional aggregates (rods)
Consider a linear aggregate where all the N particles/molecules are joined together in a line.
Most particles have 2 bonds which we will say results in a negative change in the free energy μBulk. The two particles on the end each have an exposed surface with one bond each that is unsatisfied. This is equivalent to one particle with two unsatisfied bonds. One way to write the energy change down is to firstly write the free energy change of N normal particles and then add a correction for the “one particle” which has unsatisified bonds.
To calculate the change in the free energy we differentiate this expression:
Since this value is always negative adding a single molecule to a linear aggregate will always be favourable. As a result, molecules that form these kind of aggregates will tend to form very long (essentially infinite) rods or fibres.
Excess surface area
R
t
Two dimensional aggregates (disks)
The case of 2D aggregation is slightly more complicated.
As a disk-like aggregate grows the area of the surface on which the particles attach increases with the radius of the disk. The production of this excess surface area has an energy penalty associated with it.
How many molecules are required to form an aggregate of radius R? Equating the volume of the aggregate with the volume of the N molecules from which it is made and rearranging gives:
Following the same logic as for the 1D aggregate, we will work out the bulk energy of N particles but then correct for those particles at the edge of the aggregate that have unsatisfied bonds. In this case the surface correction term is related to the area of the exposed edge and we also define the surface energy per unit area γ. We then substitute for R to obtain the expression for U.
Differentiating with respect to N gives an expression for the chemical potential:
Three Dimensional aggregates (spheres)
Similarly for 3D aggregation we consider bulk and surface terms:
Work out how N is related to R à N = (4πR3/3v)
Write down energy of bulk and surface
U = UBulk + USurface = -Nµbulk + 4πR2γ
Simplify expression to remove dependence on R
µ = dU/dN
What happens as the aggregation number increases?
For a 1D aggregate we saw that the chemical potential is always negative and so aggregates grow regardless of N. However, for 2 and 3D aggregates this is not necessarily the case.
Whether this change is positive depends on the sign of the chemical potential.
A plot of the free energy shows that the gradient dU/dN is positive for N < Nc and negative above this value. This means that for 2 and 3D aggregates adding a single molecule to a small aggregate (N< Nc) will be unfavourable. Consequently, if you were to place a ready formed aggregate of N < Nc molecules in solution it would fall apart in order to decrease the free energy of the system. However, once a critical nucleus (N=Nc) of molecules has been formed it becomes favourable for the aggregate to grow. There is therefore an energy barrier to aggregation for isolated molecules in solution which must be overcome if an aggregate is to grow. The size of this critical nucleus occurs at the turning point (μ = dU/dN)
For a 2D aggregate this gives:
and for 3D:
So how do 2D and 3D aggregates start growing?
Since all aggregates start off as individual particles this raises the question as to how 2D or 3D aggregates can start growing. We have seen that to grow, a critical cluster size has to form before it becomes energetically favourable for the aggregation to continue. The formation of these clusters or nuclei can happen in one of two ways.
Homogeneous nucleation
Heterogeneous nucleation
Homogeneous nucleation is where thermal fluctuations force enough molecules together to form a cluster/nucleus of the required size. This is usually extremely unlikely and therefore takes a very long time. There is an energy barrier that must be overcome by a chance fluctuation. For a 3D aggregate this energy barrier has a height of
The probability of this energy being supplied by chance is given by Boltzmann statistics:
Thus the probability of Homogeneous nucleation is strongly dependent on the bulk cohesive energy, the size of aggregate and the temperature. Hence aggregation is faster at higher temperatures. This is often a very small probability, meaning that nucleation takes a long time or effectively never takes place.
Heterogeneous nucleation is where foreign surfaces nucleate the formation of a critical cluster size by introducing additional interactions. (Heterogeneous nucleation). These foreign surfaces can simply be dust particles, or the surfaces of a container in which the material is being held. This mechanism is usually the cause of nucleus formation in ‘dirty’ samples.
Less exposed surface area
Smaller surface term
If an aggregate forms on an impurity, it results in less exposed surface area reducing the size of the correction Usurface. This results in a smaller critical aggregate size and a lower energy barrier to nucleation. Consequently, it is easier for an aggregate to form under these conditions.
Summary of key concepts
Aggregates form when the free energy change on adding a molecule to an aggregate is negative.
The form of the free energy or chemical potential depends upon the shape of the aggregate and the aggregation number
A critical cluster size sometimes has to form before growth can occur. This is needed to overcome the contributions due to the excess surface energy of the aggregate.
This critical cluster size can be achieved via a chance thermal fluctuation (homogeneous nucleation) or through the addition of surfaces / foreign particles (heterogeneous nucleation).
13. Micelles & Membranes
Further reading
A helpful explanation of self-assembly is also covered in the book ‘Soft Condensed Matter’, although everything required will be covered in these notes.
R.A.L. Jones
QC173.458.S62 JON
Soft Condensed Matter
R.A.L. Jones, Oxford University Press, 2002, Chapter 9
Amphiphilic molecules
Amphiphiles are molecules which have a water soluble head group and a hydrophobic tail group. The head group can be anionic (negatively charged), cationic (positively charged) or uncharged. The tail group is made up of a hydrocarbon chain.
Amphiphiles are found in soaps, food stuffs, detergents, disinfectants and can be found in cosmetics and pharmaceuticals. They are also a major component of human and animal cell membranes.
When amphiphiles sit at the interface between water and oil these 2 liquids do not come into contact as much. This reduces or eliminates the energy cost of the interface. This can be used to stabilise an interface such as the oil-water emulsion of Mayonnaise.
Oil
Water
The critical micelle concentration
When amphiphilic molecules are added to water at low concentrations, they will initially disperse and some will migrate to the surface to reduce the unfavourable contact between the hydrocarbon tails and the water. The molecules exist as monomers.
At a concentration called the critical micelle concentration aggregates (or micelles) start to form in such a way that the hydrophobic tails become shielded from the aqueous environment.
The size and structure of a micelle
The free energy per molecule of amphiphiles in a micelle has a minimum at a specific aggregation number. It is easy to see why this might be the case.
When the micelles are too big, we must introduce amphiphiles into the core of the structure. The unfavourable interactions between the hydrophilic head groups of the molecules in the core and the tail groups of surrounding amphiphiles have an energy penalty associated with them.
When micelles are too small, the headgroups of the amphiphiles at the surface are unable to pack efficiently enough to prevent water from contacting the hydrophobic tail groups in the core of the micelle. This unfavourable interaction also has an energy penalty associated with it.
As a result of these interactions, the micelles that form in solution will tend to have a well defined size which corresponds to the case when the amphililes are just able to pack together to ensure a) that no additional amphiphiles are introduced into the core of the micelle and b ) that the headgroups pack efficiently enough to shield the hydrophobic interior from water.
Optimum headgroup area
There is an optimum headgroup area, a0, occupied by an amphiphile at the interface. This is determined by a balance between competing contributions to the free energy. These contributions dominate at different head group areas.
a < a0: Electrostatic or steric repulsion between the head groups act to force neighbouring molecules apart.
a > a0: If the head groups are separated too much then hydrocarbon tails are exposed to water resulting in hydrophobic interactions.
The hydrocarbon volume and critical chain length
In addition to the headgroup area the amphiphile is characterised by two other parameters.
The hydrocarbon volume, v, is defined as the volume that is occupied by the hydrocarbon tails of the molecules.
The critical chain length, lc , is the length of the hydrocarbon chain when it is fully extended.
[It is important to realise that because of the definition of lc as the maximum length of the amphiphile tail then it is not generally true that a0*lc = v]
Factors affecting the shape of micelles
Amphiphiles are capable of forming many different shaped micelles including spheres, cylinders and bi-layers. The fluid like nature of these molecules means that we can determine the shape of micelles by considering how the amphiphiles pack together.
To do this we just require 3 parameters:
The hydrocarbon volume, v, The critical chain length, lc The optimum head group area, ao
How do these parameters determine the shape of a micelle?
The critical chain length, lc , determines the size and shape of a micelle as it sets the maximum length to which an individual amphiphile molecule can be stretched
Clearly the radius of any micelle that forms has to be less than or equal to this chain length.
Critical packing parameter
The shape of a micelle depends on the critical packing parameter.
In each case we can calculate the conditions through applying the same steps:
Calculate the number of amphiphiles in the micelle
N = surface area of micelle / a0
N = volume of micelle / v
Eliminate N to obtain an expression for R the radius of the micelle in terms of v and a0
Substitute R ≤ lc
Rearrange to find the condition on the critical packing parameter
Spherical micelles
The volume and surface area of a spherical micelle can be related to the aggregation number, N, the hydrocarbon volume, v, and the optimum head group area, ao
If the radius of the micelle is less than the critical chain length, lc for spheres to form we require that
Cylinders
Similarly for cylindrical micelles
The 1/3 comes from the fact that the spherical micelle (above) is more energetically favourable than the cylinder, since the cylinder has exposed ends with hydrophobic tails. However, when H > 1/3 it is no longer possible to form spheres.
Bi-layers
Similarly for bi-layers:
Where in a similar way the ½ originates from the fact that when H is ≤ ½ the amphiphiles will form cylinders.
Summary of key concepts
Amphiphilic molecules contain a hydrophobic head group and hydrophobic tail group.
When added to water they form micelles above a critical concentration. Their shape is determined by the volume and length of the tail and the optimum area of the molecular headgroups.
A geometric packing parameter can be used to identify whether spherical, cylindrical or bilayer structures will form.
Vesicle formation
When lipid bi-layers are formed in solution, there is an excess energy associated with the exposed hydrophobic tail groups at the edges of the structure.
The bi-layers can offset this energy by folding around to close themselves off and form an isolated shell or vesicle.
vesicle
Vesicles are closed structures that can be used to encapsulate materials in their interior for use in drug delivery. If the outside of the vesicle is decorated with specific chemical receptors which cause the vesicle to rupture and deliver its cargo when it reaches its target, it can act like a ‘magic bullet’.
Vesicles are also the most basic model for understanding the physics of the cell wall. A cell membrane consists of a vesicle with additional molecules such as membrane proteins distributed in the lipid bi-layer (A lipid is a type of amphiphile).
Bi-layer / membrane elasticity
a0
a0+Δa
r
d
Earlier we considered the optimum headgroup area a0 which is at the minimum of the free energy. However, when a bi-layer is bent the amphiphiles on the outside of the bi-layer move slightly further apart, whilst the molecules on the inside are pushed closer together. Both these changes will increase the free energy of the bilayer leading to a resistive force or elasticity.
In the simplest approximation:
Calculation: Estimating the elasticity of a membrane
Calculate the head group area at the middle and outside edge of the curved membrane:
=
Solve to find:
The number of headgroups per unit area is 2/a0. One on outer edge, one inner edge so the energy per unit area is 0.5kΔa2*2/a0. Substituting Δa gives:
where
This implies that a membrane has an elasticity with spring constant κ. It also shows that a membrane gets increasingly stiff if the constituent amphiphiles are long and / or fat.
Membrane (Entropic Repulsion)
The stiffness or bending modulus of a typical membrane is sufficiently small that they may undergo thermal induced fluctuations. Below we compare a membrane with some other typical “soft” materials.
Material |
Modulus (kPa) |
|---|---|
Rubber |
1000 |
Jelly |
10 |
Membrane |
0.1 |
The closer two membranes come together the more these “thermal undulations” become restricted by the presence of the other membrane. In comparison with the lecture on steric interactions we may be able to guess that as the number of configurations (or undulations) is reduced the entropy of the system increases. Since this is unfavourable we expect a repulsive entropic force between the membranes.
Calculation: What is the force experienced by 2 membranes?
To make this calculation tractable we have to make a number of simplifications. These it turns out affect the final prefactor but give rise to the correct physics.
We start with the following assumptions:
2 bi-layers a distance D apart will interact when the modes have amplitude D/2
From the equipartition principle each mode has kBT/2 energy associated with it.
Each mode is a 2D wave and occupies an area πx2
Comes from geometry. Write x in terms of R and D à x2≈RD
=
Use earlier result to find an alternative expression for R. ΔU = kBT/2 and A = πx2
à
The final result is:
This pressure is always repulsive and is stronger for more flexible membranes. The pressure also varies as 1/D3. This is interesting since this is the same dependence we observed for the Van der Waals interaction between two surfaces but with the opposite sign. The combined pressure would therefore be:
This has an interesting consequence that the pressure for a membrane is either always positive or always negative regardless of distance D, depending on the sign of the bracket. Compare this with the DLVO potential for a charged colloid in a solution of ions which has the electrostatic (osmotic) and VdWs terms which exhibits a maximum, meaning that at small separations the particles become stuck together, whereas at large separations the pressure is repulsive.
DLVO
Undulation + VdW
Electrostatic
Undulation
Dispersion